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Mirrors > Home > ILE Home > Th. List > nummul1c | GIF version |
Description: The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
nummul1c.1 | ⊢ 𝑇 ∈ ℕ0 |
nummul1c.2 | ⊢ 𝑃 ∈ ℕ0 |
nummul1c.3 | ⊢ 𝐴 ∈ ℕ0 |
nummul1c.4 | ⊢ 𝐵 ∈ ℕ0 |
nummul1c.5 | ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝐵) |
nummul1c.6 | ⊢ 𝐷 ∈ ℕ0 |
nummul1c.7 | ⊢ 𝐸 ∈ ℕ0 |
nummul1c.8 | ⊢ ((𝐴 · 𝑃) + 𝐸) = 𝐶 |
nummul1c.9 | ⊢ (𝐵 · 𝑃) = ((𝑇 · 𝐸) + 𝐷) |
Ref | Expression |
---|---|
nummul1c | ⊢ (𝑁 · 𝑃) = ((𝑇 · 𝐶) + 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nummul1c.5 | . . . 4 ⊢ 𝑁 = ((𝑇 · 𝐴) + 𝐵) | |
2 | nummul1c.1 | . . . . 5 ⊢ 𝑇 ∈ ℕ0 | |
3 | nummul1c.3 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
4 | nummul1c.4 | . . . . 5 ⊢ 𝐵 ∈ ℕ0 | |
5 | 2, 3, 4 | numcl 9308 | . . . 4 ⊢ ((𝑇 · 𝐴) + 𝐵) ∈ ℕ0 |
6 | 1, 5 | eqeltri 2230 | . . 3 ⊢ 𝑁 ∈ ℕ0 |
7 | nummul1c.2 | . . 3 ⊢ 𝑃 ∈ ℕ0 | |
8 | 6, 7 | num0u 9306 | . 2 ⊢ (𝑁 · 𝑃) = ((𝑁 · 𝑃) + 0) |
9 | 0nn0 9106 | . . 3 ⊢ 0 ∈ ℕ0 | |
10 | 2, 9 | num0h 9307 | . . 3 ⊢ 0 = ((𝑇 · 0) + 0) |
11 | nummul1c.6 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
12 | nummul1c.7 | . . 3 ⊢ 𝐸 ∈ ℕ0 | |
13 | 12 | nn0cni 9103 | . . . . . 6 ⊢ 𝐸 ∈ ℂ |
14 | 13 | addid2i 8019 | . . . . 5 ⊢ (0 + 𝐸) = 𝐸 |
15 | 14 | oveq2i 5836 | . . . 4 ⊢ ((𝐴 · 𝑃) + (0 + 𝐸)) = ((𝐴 · 𝑃) + 𝐸) |
16 | nummul1c.8 | . . . 4 ⊢ ((𝐴 · 𝑃) + 𝐸) = 𝐶 | |
17 | 15, 16 | eqtri 2178 | . . 3 ⊢ ((𝐴 · 𝑃) + (0 + 𝐸)) = 𝐶 |
18 | 4, 7 | num0u 9306 | . . . 4 ⊢ (𝐵 · 𝑃) = ((𝐵 · 𝑃) + 0) |
19 | nummul1c.9 | . . . 4 ⊢ (𝐵 · 𝑃) = ((𝑇 · 𝐸) + 𝐷) | |
20 | 18, 19 | eqtr3i 2180 | . . 3 ⊢ ((𝐵 · 𝑃) + 0) = ((𝑇 · 𝐸) + 𝐷) |
21 | 2, 3, 4, 9, 9, 1, 10, 7, 11, 12, 17, 20 | nummac 9340 | . 2 ⊢ ((𝑁 · 𝑃) + 0) = ((𝑇 · 𝐶) + 𝐷) |
22 | 8, 21 | eqtri 2178 | 1 ⊢ (𝑁 · 𝑃) = ((𝑇 · 𝐶) + 𝐷) |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 ∈ wcel 2128 (class class class)co 5825 0cc0 7733 + caddc 7736 · cmul 7738 ℕ0cn0 9091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-addcom 7833 ax-mulcom 7834 ax-addass 7835 ax-mulass 7836 ax-distr 7837 ax-i2m1 7838 ax-1rid 7840 ax-0id 7841 ax-rnegex 7842 ax-cnre 7844 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-br 3967 df-opab 4027 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-iota 5136 df-fun 5173 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-sub 8049 df-inn 8835 df-n0 9092 |
This theorem is referenced by: nummul2c 9345 decmul1 9359 decmul1c 9360 |
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